Empirical Bayes When Estimation Precision Predicts Parameters

Abstract

Gaussian empirical Bayes methods usually maintain aprecision independenceassumption: The unknown parameters of interest are independent from the known standard errors of the estimates. This assumption is often theoretically questionable and empirically rejected. This paper proposes to model the conditional distribution of the parameter given the standard errors as a flexibly parameterized location-scale family of distributions, leading to a family of methods that we callclose. Thecloseframework unifies and generalizes several proposals under precision dependence. We argue that the most flexible member of theclosefamily is a minimalist and computationally efficient default for accounting for precision dependence. We analyze this method and show that it is competitive in terms of the regret of subsequent decision rules. Empirically, usingcloseleads to sizable gains for selecting high-mobility Census tracts.